The last temptation of William T. Tutte
Introduction
In [3], Tutte asked if there existed a triangulation of the plane whose 4-coloring complex had more than one component of the same parity. He based this question on numerous examples he examined over the course of many years. One obvious possibility is the icosahedron, whose 4-colorings form ten Kempe equivalence classes. However, Tutte found out that the icosahedron has a connected coloring complex. Nonetheless, he was resistant to making this question into a conjecture that no such examples exist, since, in his view, “the data are too few to justify a Conjecture”. However, he asked: “If anyone knows of any case of two components of the same parity, I would be glad to hear of it”.
In this short note, we provide examples of 4-connected triangulations of the plane whose 4-coloring complexes have arbitrarily many components with odd colorings and arbitrarily many components with even colorings. Although this seems to resolve the Tutte problem, we end up with a closely related conjecture, which is based on an extensive computation, and which claims that for every planar triangulation whose 4-coloring complex is disconnected has a component of even parity and one of odd parity (see Conjecture 8).
Section snippets
Coloring complexes and the parity of 4-colorings
Given a 4-colorable graph , an independent set of vertices is said to be a color class if it is one of the color classes for some 4-coloring of . The -coloring complex is the graph which has all the color classes of as its vertices and two vertices joined by an edge if the color classes and appear together in a 4-coloring of . These graphs were introduced by Tutte [2], who discussed their basic structure. See Fig. 2 for an example of a coloring complex.
A
4-colorings of triangulated surfaces
Fisk [1] introduced an alternative view of the parity of 4-colorings. We show how to reconcile this approach with Tutte’s definition.
Given a triangulation of an orientable surface with one of its orientations fixed, we can view a 4-coloring of as a simplicial mapping onto the boundary of the tetrahedron, which will be denoted as . For each triangle of , we consider the facial triangles in that are mapped onto in such a way that their orientation is preserved and those
Coloring complexes with many components
We have found two triangulations of the plane on 12 vertices which answer Tutte’s question in the affirmative. Their 4-coloring complexes each have three components, from which it follows that two of these components must have the same parity. These triangulations are drawn in Fig. 1.
We will now determine the 4-coloring complex of Example 1, and show that it has 3 connected components (the argument for Example 2 is similar).
Theorem 4 The 4-coloring complex of the first example given in Fig. 1(a) is the
4-connected examples
Identification over a triangle produces triangulations that are not 4-connected. However, it may be that in the back of Tutte’s mind was an implicit condition that the triangulations should be4-connected. The following generalized examples show that one can also find 4-connected examples whose coloring complexes have many components (of the same parity). The two graphs and in Fig. 1 are just smallest examples in two infinite families of 4-connected triangulations. We define as
Next steps
At present, we have not found 5-connected triangulations of the plane whose 4-coloring complexes would have arbitrarily many components. However, we have found some 5-connected examples whose 4-coloring complexes have more than one component of the same parity. The three smallest triangulations of this kind are illustrated in Fig. 4. These are triangulations of the plane on 20 vertices, and their minimality follows from an extensive computation that we performed on 5-connected triangulations
References (3)
Geometric coloring theory
Adv. Math.
(1977)
Cited by (0)
- 1
Supported in part by an NSERC Discovery Grant R611450 (Canada), by the Canada Research Chair program, and by the Research Grant J1-8130 of ARRS (Slovenia) .
- 2
On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.